Quantum computer and quantum computing method

ABSTRACT

According to an embodiment, a quantum computer includes physical systems X i , a physical system Y j  and a light source unit. The physical systems X i  and the physical system Y j  are provided in a cavity. Each physical system X i  includes states |0&gt; i , |1&gt; i , |2&gt; i  and |e&gt; i , the states |0&gt; i  and |1&gt; i  being used for a qubit, a |2&gt; i -|e&gt; i  transition being resonant with a cavity mode of the cavity. The physical system Y j  includes states |2&gt;′ j  and |e&gt;′ j , a |2&gt;′ j -|e&gt;′ j  transition being resonant with the cavity mode. The light source unit applies laser beams to the cavity to manipulate states of two of physical systems X i , the laser beams including a laser beam for collecting population in the state |2&gt;′ j  in the |2&gt;′ j -|e&gt;′ j  transition.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a Continuation Application of PCT Application No.PCT/JP2014/081415, filed Nov. 27, 2014 and based upon and claiming thebenefit of priority from Japanese Patent Application No. 2014-006049,filed Jan. 16, 2014, the entire contents of all of which areincorporated herein by reference.

FIELD

Embodiments described herein relate to a quantum computer utilizing thecoupling between a cavity and a physical system.

BACKGROUND

In recent years, researches are being made of a quantum computer whichperforms a computation by using quantum-mechanical superposition. As oneof quantum computers, a quantum computer based on frequency domainquantum computation that discriminates between qubits (quantum bits) infrequency domains is known in the art. In the frequency domain quantumcomputation, qubits are not discriminated in terms of their positions.Therefore, even qubits that are not to be manipulated are under theeffect of operation light with detuning, causing undesired interaction.The undesired interaction may cause gate errors. The adverse effectscaused by the undesired interaction may decrease if the frequencydifference of the transitions used for qubits is very large. Where thetransition distributed in a finite frequency domain is used, however,transitions of small frequency differences may have to be used to load alarge number of qubits. Even where the transitions of small frequencydifferences are used, a quantum computer required is a computer based onthe frequency domain quantum computation capable of performing a quantumgate while suppressing the effects caused by the undesired interaction.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 illustrates frequency domain quantum computation.

FIG. 2 illustrates undesired interaction in the physical systems usedfor the frequency domain quantum computation.

FIG. 3 illustrates a quantum computer according to an embodiment.

FIG. 4 illustrates a physical system which is used for the frequencydomain quantum computation in the embodiment.

FIG. 5 illustrates part of the energy states of the Pr³⁺ ion in theY₂SiO₅ used in the embodiment.

FIG. 6 is a flowchart illustrating a method for manipulating twophysical systems according to the embodiment.

FIG. 7 is a waveform chart illustrating pulses used for the adiabaticpassage via a cavity according to the embodiment.

FIG. 8 illustrates a quantum computer according to another embodiment.

FIG. 9 illustrates a quantum computer according to still anotherembodiment.

DETAILED DESCRIPTION

According to one embodiment, a quantum computer includes physicalsystems X_(i) (i=1, 2, . . . , N; N being an integer not less than 2), aphysical system Y_(j) (j=1, 2, . . . , N₂; N₂ being an integer not lessthan 1), and a light source unit. The physical systems X, are providedin a cavity, each of the physical systems X_(i) comprising at least fourstates including states |0>_(i), |1>_(i), |2>_(i) and |e>_(i), energy ofthe state |e>_(i) being higher than energy of the states |0>_(i) and|1>_(i) used for a qubit and energy of the state |2>_(i) used forassisting a gate operation, a |2>-|e>_(i) transition being resonant witha cavity mode of the cavity. The physical system Y_(j) is provided inthe cavity, the physical system Y_(j) being different from the physicalsystems X_(i), the physical system Y_(j) comprising at least two energystates including states |2>′_(j) and |e>′_(j), energy of the state|e>′_(j) being higher than energy of the state |2>′_(j), a|2>′_(j)-|e>′_(j) transition being resonant with the cavity mode. Thelight source unit applies laser beams to the cavity to manipulate astate of a physical system X_(a) (where s is a natural number not morethan N) and a state of physical system X_(t) (where t is a naturalnumber not more than N and different from s) of the physical systemsX_(i), the laser beams including a first laser beam resonating with a|1>_(s)-|e>_(s) transition, a second laser beam resonating with a|1>_(t)-|e>_(t) transition, and a third laser beam for collecting,population in the state |2>′_(j) in the |2>′_(i)-|e>′_(j) transition ofthe physical system Y_(j).

Hereinafter, various embodiments will be described with reference to thedrawings. In the following embodiments, the like elements will bedenoted by the like reference symbols, and redundant descriptions willbe omitted where appropriate.

First of all, a description will be given of frequency domain quantumcomputation and “undesired interaction” which may occur at the quantumgate thereof. A description will also be given of the “resonancecondition” under which the effects of the undesired interaction areparticularly marked. Further, a description will be given of a quantumgate operation method and configuration, which control the resonancecondition so as to eliminate the effects of the undesired interaction.

[Frequency Domain Quantum Computation]

In the frequency domain quantum computation, a plurality of physicalsystems, which are arranged in an optical cavity (also called an opticalresonator) and each of which has a transition resonating with a commoncavity mode (eigenmode) and has other transitions different in frequencydepending upon the physical systems, are used as qubits. For example,ions and atoms can be used as the physical systems. In the frequencydomain quantum computation, each physical system can be selectivelymanipulated by radiating a laser beam resonating with the transitionfrequency of the physical system.

A description will be given of the case where N four-level systems X_(i)(i=1, 2, 3, . . . , N) are used as physical systems. It is noted herethat N is an integer which is not less than 2. Each of the four-levelsystems X_(i) has four energy states. The four energy state will beexpressed as |0>_(i), |1>_(i), |2>_(i), and |e>_(i) in the energy levelascending order. The suffix i attached to each state (each ket vector)is for identifying the four-level systems X_(i). In the following, thesuffix i may be omitted. The states |0>_(i) and |1>_(i) are used for aqubit, and the state |2>_(i) is used for assisting the gate operation.The excited state |e>_(i) has a higher level of energy than the states|0>_(i), |1>_(i) and |2>_(i). The |2>_(i)-|e>_(i) transition (namely, atransition between the state |2>_(i) and the state |e>_(i)) is atransition resonating with a common cavity mode of the cavity. Thefrequency of the |1>_(i)-|e>_(i) transition differs depending upon thefour-level systems X_(i).

FIG. 1 illustrates a physical system that is used for the frequencydomain quantum computation when N=3. As shown in FIG. 1, physicalsystems X₁, X₂ and X₃ are coupled to a common cavity mode. To bespecific, the |2>_(i)-|e>₁ transition, the |2>₂-|e>₂ transition and the|2>₃-|e>₃ transition resonate with the common cavity mode. That is, thefrequency of the |2>₁-|e>₁ transition, the frequency of the |2>₂-|e>₂transition and the frequency of the |2>₃-|e>₃ transition are equal tothe resonance frequency of the common cavity mode. The frequency of the|1>₁-|e>₁ transition, the frequency of the |1>₂-|e>₂ transition and thefrequency of the |1>₃-|e>₃ transition differ from each other.

Where two of these four-level systems X_(i) are manipulated (forexample, the states of four-level systems X₁ and X₂ are manipulated),operation light L₁ resonating with the |1>₁-|e>₁ transition andoperation light L₂ resonating with the |1>₂-|e>₂ transition are radiatedto all four-level systems X_(i). Ideally, the states of four-levelsystems X₁ and X₂ can be selectively manipulated by the radiation ofoperation light L1 and operation light L2. As a method for manipulatingthe states used in a quantum gate, an adiabatic passage via a cavity isknown in the art. In this method, when, for example, the state of thefour-level system X₁ is manipulated from the initial state |1>₁ to thestate |2>₁ and the state of the four-level system X₂ is manipulated fromthe initial state |2>₂ to the state |1>₂, the intensities of theoperation light L₁ and L₂ are controlled in such a manner that the Rabifrequencies Ω₁ and Ω2 of the operation light L₁ and L₂ satisfyExpression (1) below under the condition where τ₁>τ₂.

$\begin{matrix}{{{\Omega_{1}(t)} = {\Omega_{0}{\exp\left\lbrack {- \frac{\left( {t - \tau_{1}} \right)^{2}}{2\sigma^{2}}} \right\rbrack}}}{{\Omega_{2}(t)} = {\Omega_{0}{\exp\left\lbrack {- \frac{\left( {t - \tau_{2}} \right)^{2}}{2\sigma^{2}}} \right\rbrack}}}} & (1)\end{matrix}$

[Undesired Interactions]

Strictly speaking, undesired interactions described below inevitablyoccur in the frequency domain quantum computation. Since the four-levelsystems X_(i) are not discriminated from one another in light of theirpositions, operation light with detuning is applied to four-levelsystems X_(i) which are not to be manipulated, leading to the undesiredinteractions. To be specific, the undesired interactions includeinteractions caused by operation light L₁ in four-level systems X₂ andX_(j) (j=3, 4, . . . , N) and interactions caused by operation light L₂in four-level systems X₁ and X_(j) (j=3, 4, . . . , N). FIG. 2illustrates a physical system including undesired interactions when N=3.In FIG. 2, the solid-line two-headed arrow connecting the state |l>_(i)and the state |e>_(i) indicates a necessary interaction, and abroken-line two-headed arrow indicates an undesired interaction. Thethick solid-line two-headed arrow connecting the state |2>_(i) and thestate |e>_(i) indicates a cavity mode.

If the frequency difference between the |1>_(i)-|e>_(i) transitions ofthe four-level systems X_(i) is very large, the undesired interactiondoes not cause a significant effect. However, since the frequencies ofthe transitions are distributed in a finite frequency domain,transitions of a small frequency difference should be utilized to use alarge number of qubits. For this reason, it is desirable that thequantum gate can be performed with high efficiency even if the frequencydifferences are small.

[Resonance Condition]

A description will be given of the property of an undesired interaction.The Hamiltonian describing a physical system used for the frequencydomain quantum computation including an undesired interaction (such asthe physical system depicted in FIG. 2) is expressed by Expression (2)set forth below.

$\begin{matrix}{{{H(t)}/\hslash} = {{\sum\limits_{i = 1}^{N}\;\left\lbrack {{\sum\limits_{{j = 0},1,2,e}\;{\omega_{j}^{(i)}\sigma_{jj}^{(i)}}} - {i\;{\gamma\sigma}_{ee}^{(i)}}} \right\rbrack} + \left\lbrack {{2\pi\; f_{g}a^{\dagger}a} - {i\;\kappa\; a^{\dagger}a}} \right\rbrack + {\sum\limits_{i = 1}^{N}\;\left\lbrack {{{ga}\;\sigma_{e\; 2}^{(i)}} + {{\Omega_{1}(t)}e^{{- i}\; 2\;\pi\; f_{1}t}\sigma_{e\; 1}^{(i)}} + {{\Omega_{2}(t)}e^{{- i}\; 2\;\pi\; f_{2}t}\sigma_{e\; 1}^{(i)}} + {H.c.}} \right\rbrack}}} & (2)\end{matrix}$where σ_(ab) ^((i)) is an operator for transition of state |b>_(i) offour-level system X_(i) to state |a>_(i), a and

are the annihilation operator and the creation operator of the cavitymode, respectively, g is a coupling constant for coupling between thecavity mode and a physical system, γ is a relaxation rate of transition,κ is an attenuation constant of the cavity, and H.c. is a Hermitianconjugate.

The terms of the Hamiltonian will be described. The first term includesan energy term of each state of each ion and an energy relaxation termof each ion. The second term includes an energy term and a cavityrelaxation term of the cavity mode. In the following, a description willbe given of the case where γ=κ=0 on the assumption of an idealstrongly-coupled system. The third term is an interaction term,including an interaction between the cavity mode and the |2>_(i)-|e>_(i)transition of each four-level system X_(i), an interaction between theoperation light L₁ and the |1>_(i)-|e>_(i) transition of each four-levelsystem X_(i), and an interaction between the operation light L₂ and the|1>_(i)-|e>_(i) transition of each four-level system X_(i).

The interaction Hamiltonian in the Hamiltonian given by Expression (2)is expressed as Expression (4) below, in which H₀ given by Expression(3) below is used.

$\begin{matrix}{{{H_{0}/\hslash} \equiv {{\sum\limits_{i = 1}^{N}\;\left\lbrack {\sum\limits_{{j = 0},1,2,e}\;\left( {\omega_{j}^{(i)}\sigma_{jj}^{(i)}} \right)} \right\rbrack} + {2\pi\; f_{g}a^{\dagger}a}}}\begin{matrix}{{{H^{\prime}(t)}/\hslash} = {{\exp\left\lbrack {{- i}\; H_{0}{t/\hslash}} \right\rbrack}\left( {H/\hslash} \right){\exp\left\lbrack {i\; H_{0}{t/\hslash}} \right\rbrack}}} \\{= {{\sum\limits_{i = 1}^{N}\;{{ga}\;\sigma_{e\; 2}^{(i)}}} + {{\Omega_{1}(t)}\left\{ {\sigma_{e\; 1}^{(1)} + {e^{{- i}\;\Delta\; t}\sigma_{e\; 1}^{(2)}} + {\sum\limits_{j = 3}^{N}\;{e^{{- i}\;\Delta_{j}t}\sigma_{e\; 1}^{(j)}}}} \right\}} +}} \\{{{\Omega_{2}(t)}\left\{ {{e^{i\;\Delta\; t}\sigma_{e\; 1}^{(1)}} + \sigma_{e\; 1}^{(2)} + {\sum\limits_{j = 3}^{N}\;{e^{{- {i{({\Delta + \Delta_{j}})}}}t}\sigma_{e\; 1}^{(j)}}}} \right\}} + {H.c.}}\end{matrix}} & (3)\end{matrix}$

In Expression (4), the following relations are used.(ω_(e) ⁽¹⁾−ω₁ ⁽¹⁾)−2πƒ₁=0, (ω_(e) ⁽¹⁾−ω₁ ⁽¹⁾)−2πƒ₂=Δ, (ω_(e) ⁽²⁾−ω₁⁽²⁾)−2πƒ₁=−Δ,(ω_(e) ⁽²⁾−ω₁ ⁽²⁾)−2πƒ₂=0, (ω_(e) ^((j))−ω₁ ^((j)))−2πƒ₁=Δ_(j), (ω_(e)^((j))−ω₁ ^((j)))−2πƒ₂=Δ+Δ_(j),(ω_(e) ^((i))−ω₂ ^((i))−)2πƒ_(g)=0, for i=1,2, . . . N and j=3,4, . . ., N.

Δ is a frequency difference between the |e>₁-|1>_(j) transition and the|e>₂-|1>₂ transition, and Δ_(j) is a frequency difference between the|e>₁-|1>₁ transition and the |e>_(j)-|1>_(j) transition.

The interaction Hamiltonian given by Expression (4) is divided into anecessary interaction H₁ and an undesired interaction V.

$\begin{matrix}{{H^{\prime}(t)} = {{H_{1}(t)} + {{V(t)}\left\{ \begin{matrix}{{{H_{1}(t)}/\hslash} = {{\sum\limits_{i = 1}^{N}\;{{ga}\;\sigma_{e\; 2}^{(i)}}} + {{\Omega_{1}(t)}\sigma_{e\; 1}^{(1)}} + {{\Omega_{2}(t)}\sigma_{e\; 1}^{(2)}} + {H.c.}}} \\{{{V(t)}/\hslash} = {{{\Omega_{1}(t)}\left\{ {{e^{{- i}\;\Delta\; t}\sigma_{e\; 1}^{(2)}} + {\sum\limits_{j = 3}^{N}\;{e^{{- i}\;\Delta_{j}t}\sigma_{e\; 1}^{(j)}}}} \right\}} +}} \\{{{\Omega_{2}(t)}\left\{ {{e^{i\;\Delta\; t}\sigma_{e\; 1}^{(1)}} + {\sum\limits_{j = 3}^{N}\;{e^{{- {i{({\Delta + \Delta_{j}})}}}t}\sigma_{e\; 1}^{(j)}}}} \right\}} + {H.c.}}\end{matrix} \right.}}} & (5)\end{matrix}$

In the adiabatic passage via the cavity, the error probability caused bythe undesired interactions can be calculated by perturbation theory,using Expression (5). An initial state |ψ⁽0)

is assumed to be a dark state |ψ0⁽0)

that is one of eigenstates of H₁. The error probability in the adiabaticpassage is the probability of transition to another eigenstate |ψ_(n)^((t))

^((n≠)0) at time t. In order to perform this manipulation with highefficiency, it is general to use operation light L₁ and operation lightL₂ that satisfy Ω_(1,2)<<g. Since V is smaller than H₁ in this case, thetime evolution can be calculated by performing perturbation theory withrespect to V, as shown in Expression (6) set forth below.

$\begin{matrix}{{\left. {\left. {\left. {\left. \left| {\psi(t)} \right. \right\rangle = \left| {\psi_{0}(t)} \right.} \right\rangle + {\sum\limits_{n}\; C_{n}^{(1)}}} \middle| {\psi_{n}(t)} \right\rangle + {\sum\limits_{n}\; C_{n}^{(2)}}} \middle| {\psi_{n}(t)} \right\rangle + {O\left( V^{3} \right)}}\left\{ \begin{matrix}{{C_{n}^{(1)}(t)} = {\frac{1}{i\;\hslash}{\int_{0}^{t}\ {d\; t^{\prime}e^{{- {({E_{n} - E_{0}})}}{t^{\prime}/i}\;\hslash}\left\langle {{\psi_{n}\left( t^{\prime} \right)}{{V\left( t^{\prime} \right)}}{\psi_{0}\left( t^{\prime} \right)}} \right\rangle}}}} \\{{C_{n}^{(2)}(t)} = {\left( \frac{1}{i\;\hslash} \right)^{2}{\int_{0}^{t}\ {d\; t^{\prime}{\int_{0}^{t^{\prime}}\ {d\; t^{''}e^{{- {({E_{n} - E_{k}})}}{t^{\prime}/i}\;\hslash}e^{{- {({E_{k} - E_{0}})}}{t^{\prime}/i}\;\hslash}}}}}}} \\{\left\langle {\psi_{n}\left( t^{\prime} \right)} \middle| {V\left( t^{\prime} \right)} \middle| {\psi_{k}\left( t^{\prime} \right)} \right\rangle\left\langle {{\psi_{k}\left( t^{''} \right)}{{V\left( t^{''} \right)}}{\psi_{0}\left( t^{''} \right)}} \right\rangle}\end{matrix} \right.} & (6)\end{matrix}$

In Expression (6), E_(n) is an eigenvalue corresponding to theeigenstate |ψ_(n)

of H₁. E_(n) changes with time in accordance with Ω_(1,2) alone. In thecase where Ω_(1,2)<<g, changes in E_(n) are sufficiently small, comparedwith the absolute value of E_(n). Therefore, when the exponents of theexponent functions in coefficients C_(n) ⁽¹⁾ and C_(n) ⁽²⁾ are zero, theerror probability increases to such an extent that the perturbationtheory becomes invalid. In view of the fact that V(t′) contains anoscillation term, the resonance condition given by Expression (7) belowcan be obtained from the first-order coefficient C_(n) ⁽¹⁾ as acondition under which the error probability of the adiabatic passage viathe cavity increases.(E _(n) −E ₀)/

=±Δ,±Δ_(j),±(Δ+Δ_(j))  (7)

Likewise, the resonance condition given by Expression (8) can beobtained from the second-order coefficient.

$\begin{matrix}\left\{ \begin{matrix}{{{\left( {E_{k} - E_{0}} \right)/\hslash} = {\pm \Delta}},{\pm \Delta_{j}},{\pm \left( {\Delta + \Delta_{j}} \right)}} \\{{{\left( {E_{n} - E_{k}} \right)/\hslash} = {\pm \Delta}},{\pm \Delta_{j}},{\pm \left( {\Delta + \Delta_{j}} \right)}}\end{matrix} \right. & (8)\end{matrix}$

Compared with the resonance condition obtained from the first-ordercoefficient C_(n) ⁽¹⁾, the resonance condition obtained from thesecond-order coefficient C_(n) ⁽²⁾ does not contribute much. However,when the two or more conditions are satisfied simultaneously, thecontribution is significantly large. As a condition for permitting twoor more conditions to be satisfied in the resonance condition ofExpression (8), the following condition given by Expression (9) isobtained.(E _(n) −E ₀)/

=±Δ,±Δ_(j),±2Δ,±2Δ_(j),±(Δ−Δ_(j)),±(2Δ+Δ_(j)),±(Δ+2Δ_(j)),±2(Δ+Δ_(j))  (9)for j=3, 4, . . . , N.

The analytical solution to the resonance condition can be obtained bycalculating the eigenvalue E_(n) of H₁. Where the term of Ω_(1,2) of H₁is set at zero for the case where Ω_(1,2)<<g, the eigenvalue E_(n) canbe obtained based on the analogy of well-known vacuum Rabi splitting.The number of four-level systems X_(i) having a population in state|2>_(i) or state |e>_(i) will be denoted by N₂, the number of four-levelsystems X_(i) having a population in state |e>_(i) will be denoted byn_(e), and the number of photons in the cavity mode will be denoted byn_(c). The eigenvalues of H₁ can be classified by designating the totalnumber N_(e) of excitons (N_(e)=n_(e)+n_(c)) and a maximal valuen_(e)|_(max) of n_(e). Part of the eigenvalues of H₁ are given byExpression (10) below.E _(1,N) ₂ /

=±√{square root over (N ₂)}g(N _(e)=1,n _(e)|_(max) =N ₂)E _(2,N) ₂ /

=0,±√{square root over (4N ₂=2)}g(N _(e)=2,n _(e)|_(max) =N ₂)E _(1,N) ₂ ⁻¹/

=±√{square root over (N ₂−2)}g(N _(e)=2,n _(e)|_(max) =N ₂−1)  (10)

By using Expressions (7), (9) and (10), the resonance condition underwhich undesired interactions increase can be analytically obtained. Theresonance condition should be avoided in order to perform the adiabaticpassage via the cavity with high efficiency, namely, to perform thequantum gate efficiently in the frequency domain quantum computation.

[Control of Resonance Condition]

As shown in Expression (10), the eigenvalues of H₁ vary in accordancewith the number N₂ of four-level systems X_(i) having a population ineither state |2>_(i) or state |e>_(i).

This means that the resonance condition can be controlled based on thenumber of four-level systems X_(i) having a probability amplitude in thetransition coupled to the cavity. This kind of control is referred to asa control based on the addition of the transition coupled to the cavity.

For example, in a quantum computer utilizing three qubits, the adiabaticpassage via the cavity is performed such that four-level systems X₁, X₂and X₃ are manipulated from the initial states |1>₁, |2>₂ and |1>₃ tostates |2>₁, |1>₂ and |1>₃. In the manipulation, where N_(e) is equal to1, N₂ is equal to 2, and where N_(e) is equal to 2, N₂ is equal to 3.Therefore, the eigenvalues of H₁ can be obtained based on Expression(10), in the way expressed in Expression (11). The resonance conditioncan be obtained from Expressions (7) and (9), using the eigenvaluesrepresented by Expression (11).E _(1,2)=±√{square root over (2)}gE _(2,3)=0,±√{square root over (10)}gE _(2,2) =±g  (11)

Where three physical systems having a transition probability in thetransition coupled to the cavity are added, the eigenvalues of H₁ can beobtained in the way indicated by Expression (12) below.E _(1,2)=±√{square root over (5)}gE _(2,3)=0,±√{square root over (22)}gE _(2,2)=±2g  (12)

In this case, the region in which the resonance condition can be avoidedsignificantly increases in the region where |Δ|<g and |A_(j)|<g.Therefore, if a frequency distribution of the four-level systems X_(i)is stochastically provided, the probability that the quantum gate can beperformed with high efficiency increases. In addition, more qubits canbe used.

As described above, the undesired interactions can be suppressed bycontrolling the resonance condition by adding a transition coupled tothe cavity, and as a result, frequency domain quantum computation can beperformed with high efficiency. Please note that what is describedherein is nothing but an example. With respect to the frequencydistribution of physical systems actually provided, the number ofadditional transitions coupled to the cavity can be changed in such amanner that control to a more appropriate condition is enabled.

Hereinafter, embodiments will be described with reference to FIGS. 3 to9.

First Embodiment

FIG. 3 schematically illustrates a quantum computer 300 according to thefirst embodiment. According to the first embodiment, Pr³⁺ ions doped inY₂SiO₅ crystal are used as specific physical systems that can beregarded as four-level system X_(i). As shown in FIG. 3, the quantumcomputer 300 includes a sample 315, which serves as a cavity whereindielectric multi-layer mirrors are arranged on sides of the Pr³⁺-iondoped Y₂SiO₅ crystal (Pr:YSO). The sample 315 is arranged inside acryostat 316 and kept at a low temperature (e.g., at 4K).

Let us consider the case where, as shown in FIG. 4, six Pr³⁺ ions areused in the frequency domain quantum computation and three of these Pr³⁺ions are used as qubits. The three Pr³⁺ ions used as qubits will beexpressed as X₁, X₂ and X₃, and the remaining three Rr³⁺ ions will beexpressed as Y₁, Y₂ and Y₃. The ions Y₁, Y₂ and Y₃ are used forcontrolling the resonance condition.

To be specific, each of the ions X₁, X₂ and X₃ has states |0>, |1>,|2>and |e> in the ascending order of energy level. The states |0>_(i)and |1>_(i) are used for the qubit, and the state |2>_(i) is used forassisting the gate operation. The |2>-|e> transitions of the ions X₁, X₂and X₃ resonate with the common cavity mode of the cavity. Each of theions Y₁, Y₂ and Y₃ has states |0>′, |1>′, 12>′ and |e>′ in the ascendingorder of energy level. In FIG. 4, illustration of the states |0>′ and|1>′of each of the ions Y₁, Y₂ and Y₃ is omitted. The |2>′-|e>′transitions of the ions Y₁, Y₂ and Y₃ resonate with the common cavitymode of the cavity. The prime “′” is attached to the states of the ionsY₁, Y₂ and Y₃. The addition of the prime is intended to mean that thephysical systems used for controlling the resonance conditions and thephysical systems used as qubits may be different in kind. The ions Y₁,Y₂ and Y₃ may be the same as the ions X₁, X₂ and X₃ (Pr³⁺ ions in thisexample). Alternatively, they may be different from the ions X₁, X₂ andX₃ in kind, or may be a combination of different kinds of ions. Wherethe ions Y₁, Y₂ and Y₃ are different in kind from the ions X₁, X₂ andX₃, the ions Y₁, Y₂ and Y₃ do not have to have state |0>′ or |1>′.

FIG. 5 illustrates part of the energy states of the Pr³⁺ ion in theY₂SiO₅ crystal. In FIG. 5, three (|±1/2>, |=3/2> and |±5/2>) of thehyperfine structure states of a ground state ³H₄ are shown, and three(|±1/2>, |=3/2> and |±5/2>) of the hyperfine structure states of anexcited state ¹D₂ are shown. The wavelength corresponding to thetransition frequency between the excited state ¹D₂ and the ground state³H₄ is approximately 606 nm. The hyperfine structure states |±5/2>,|±3/2> and |7E1/2> of the ground state ³H₄ correspond to the states |0>,|1> and |2> shown in FIG. 4, and the hyperfine structure state |5/2> ofthe excited state ¹D₂ corresponds to the state |e> shown in FIG. 4. Inthis case, the |0>-|e> transition, the |1>-|e> transition and the|2>-|e> transition are optically allowed, and the |0>-|1> transition,the |0>-|2> transition and the |1>-|2> transition are opticallyforbidden.

In the quantum computer 300 shown in FIG. 3, the ring dye laser 302excited by the argon ion laser 301 is used as a light source. A laserbeam emitted from the ring dye laser 302 is split into three laser beamsby means of two beam splitters 303 and 304, and these three laser beamsare guided to acousto-optic modulators 306, 307 and 308, respectively.To be specific, the beam splitter 303 splits the laser beam emitted fromthe ring dye laser 302 into two laser beams, and one of them enters theacousto-optic modulator 306 and the other is directed to the beamsplitter 304. The beam splitter 304 splits the incident laser beam intotwo laser beams, and one of them enters the acousto-optic modulator 307and the other is reflected by a mirror and then enters the acousto-opticmodulator 308.

The acousto-optic modulators 306, 307 and 308 modulate the incidentlaser beams in accordance with a signal generated by a controller 309,thereby generating modulated laser beams 351, 352 and 353, respectively.The modulated laser beam 351 is guided to the sample 315 by mirrors 310and 311 and lens 314. The modulated laser beam 352 is guided to thesample 315 by the lens 314. The modulated laser beam 353 is guided tothe sample 315 by mirrors 312 and 313 and the lens 314. In the presentembodiment, the argon ion laser 301, the ring dye laser 302, the beamsplitters 303 and 304, the mirror 305, the acousto-optic modulators 306to 308, the mirrors 310 to 313 and the lens 314 jointly form the lightsource unit 320.

A method for manipulating the ions X₁ and X₂, which are qubits includedin the three qubits, will be specifically described, referring to FIGS.3, 6 and 7. First, the light source unit 320 applies the modulated laserbeam 353 to the sample 315, so that a population is collected in state|2> of transition |2>-|e>, with respect to the ions Y₁, Y₂ and Y₃ (stepS601 in FIG. 6). To be specific, the acousto-optic modulator 308modulates the incident laser beam such that the modulated laser beam 353resonates with the |0>-|e> transition or the |1>-|e> transition.

Subsequently, in the state where the modulated laser beam 353 is keptapplied, the light source unit 320 simultaneously applies the modulatedlaser beams 351 and 352, which are for manipulating the ions X₁ and X₂,to the sample 315 (step S602 in FIG. 6). The “simultaneous application”is intended to mean that the application time of the modulated laserbeam 351 and the application time of the modulated laser beam 352overlap each other at least in part. To be specific, the acousto-opticmodulator 306 modulates the incident laser beam such that the Rabifrequency Ω₁ of the modulated laser beam 351 changes in the way shown inExpression (1), and the acousto-optic modulator 307 modulates theincident laser beam such that the Rabi frequency Ω₂ of the modulatedlaser beam 352 changes in the way shown in Expression (1). When thestates of the ions X₁ and X₂ are changed from the initial states |1>₁and |2>₂ to the states |2>₁ and |1>₂, τ₁ and τ₂ are made to satisfyτ₁>τ₂. FIG. 7 illustrates temporal changes in the Rabi frequencies Ω₁and Ω₂ when the ions X₁ and X₂ is manipulated from the initial states|1>₁ and |2>₂ to the states |2>₁ and |1>₂. In FIG. 7, the transverseaxis represents time, while the vertical axis represents Rabifrequencies. The waveform indicated by the dashed-dotted line is Rabifrequency Ω₁, and the waveform indicated by the solid line is Rabifrequency Ω₂. As an example, the parameters of the Rabi frequencies Ω₁and Ω₂ of the laser beams 351 and 352 are set as follows: Ω₀=1 kHz,τ₁=64.1 ms, τ₂=55.9 ms and σ→20 ms.

The modulated laser beams 351, 352 and 353 are applied to the sample 315in this manner, and this enables the ions the X₁ and X₂ to be changedfrom the initial states |1>₁ and |2>2 to the state |2>₁ and |1>₂, andyet the state of the ions X₃ (e.g., |1>₃) is kept unchanged whileavoiding the resonance condition.

As described above, the quantum computer according to the firstembodiment uses a physical system which is different from physicalsystems used as qubits and which include a transition coupling with thecommon cavity mode. By using such physical systems, the quantum gate canbe performed very efficiently, while suppressing the undesiredinteractions.

Second Embodiment

In the first embodiment, one light source is used for generating laserbeams used for state manipulations. In the second embodiment, a lightsource is prepared for each of laser beams used for state manipulations.

FIG. 8 schematically illustrates a quantum computer 800 according to thesecond embodiment. As shown in FIG. 8, the quantum computer 800 includessemiconductor lasers 801, 802 and 803. Laser beams emitted from thesemiconductor lasers 801, 802 and 803 are guided to acousto-opticmodulators 306, 307 and 308, respectively. Since the operation of theacousto-optic modulators 306, 307 and 308 is similar to that describedin the first embodiment, the explanation thereof is omitted. In thepresent embodiment, the semiconductor lasers 801 to 803, theacousto-optic modulators 306-308, the mirrors 310-313 and the lens 314jointly form a light source unit 820.

Like the first embodiment, the second embodiment enables efficientperformance of a quantum gate.

Third Embodiment

FIG. 9 schematically illustrates a quantum computer 900 according to thethird embodiment. The quantum computer 900 shown in FIG. 9 is similar inconfiguration to the quantum computer 300 of the first embodiment shownin FIG. 3. In the following, a description will be given mainly of thefeatures distinguishing the third embodiment from the first embodiment.

In the third embodiment, ions X₁, X₂ and X₃ used as qubits and ions Y₁,Y₂ and Y3 used for the control of the resonance condition are located indifferent space regions. For example, the ions X₁, X₂ and X₃ areselected from the Pr³⁺ ions in the upper portion of a sample 315, andthe ions Y₁, Y₂ and Y₃ are selected from the Pr³⁺ ions in the lowerportion of the sample 315. In this case, as shown in FIG. 9, the lightsource unit 920 applies the modulated laser beams 351 and 352 to theupper portion of the sample 315, and applies the modulated laser beam353 to the lower portion of the sample 315.

Like the first embodiment, the third embodiment enables efficientperformance of a quantum gate.

Fourth Embodiment

In connection with the fourth embodiment, a description will be given ofan example of a method for selecting Pr³⁺ ions used as qubits from thePr³⁺ ions coupling with the common cavity mode. The quantum computer ofthe fourth embodiment is similar in configuration to the quantumcomputer 300 (FIG. 3) of the first embodiment and performs the quantumgate in the same method as the first embodiment. Accordingly, arepetitive description will be omitted.

M Pr³⁺ ions included in a sample 315 and including transitions couplingwith the common cavity mode will be referred to as ion 1, ion 2, . . . ,ion M in the ascending order of the frequency of transition |1>-|e>. Itis noted here that M is an integer which is not less than 2. Thefrequency difference between the |1>₁-|e>₁ transition and the |1>₂-|e>₂transition will be expressed as Δ, and the frequency difference betweenthe |1>₁-|e>₁ transition and the |1>_(j)-|e>_(j) transition (j: aninteger not less than 3 and not more than M) will be expressed as Δ_(j).In this case, such ions as satisfy Ω₀<Δ<N₂g/N and Ω₀<Δ_(j)<N₂g/N areused as qubits. A predetermined number of ions used for the control ofthe resonance condition are selected in the descending order of thefrequency difference Δ_(j).

The number N₂ of physical systems (Pr³⁺ ions in this example) used forthe control of the resonance condition may be any number not less than1, but is preferably a value that is at least equal to the square of thenumber of qubits. This is because the eigenvalues of H₁ are practicallydependent on the square root of N₂, as shown in Expression (10). Forexample, where three qubits are used, nine Pr³⁺ ions includingtransitions coupling with the common cavity mode are used as physicalsystems for the control of the resonance condition. The nine Pr³⁺ ions(referred to as ions Y₁ to Y₉) are selected from among the ions 1 to Min the descending order of frequency difference Δ_(i) as in the methoddescribed above. In this case, manipulations similar to those performedfor the ions Y₁, Y₂ and Y₃ in the first embodiment are performed for theions Y₁ to Y₉. To be specific, the sample 315 is irradiated with a laserbeam that is modulated in such a manner as to enable the population tobe collected in state |2> with respect to the ions Y₁ to Y₉.

Like the first embodiment, the fourth embodiment enables efficientperformance of a quantum gate.

While certain embodiments have been described, these embodiments havebeen presented by way of example only, and are not intended to limit thescope of the inventions.

Indeed, the novel embodiments described herein may be embodied in avariety of other forms; furthermore, various omissions, substitutionsand changes in the form of the embodiments described herein may be madewithout departing from the spirit of the inventions. The accompanyingclaims and their equivalents are intended to cover such forms ormodifications as would fall within the scope and spirit of theinventions.

What is claimed is:
 1. A quantum computer comprising: physical systemsX_(i) (i=1, 2, . . . , N; N being an integer not less than 2) providedin a cavity, each of the physical systems X_(i) comprising at least fourstates including states |0>_(i), |1>_(i), |2>_(i) and |e>_(i), energy ofthe state |e>_(i) being higher than energy of the states |0>_(i) and|1>i used for a qubit and energy of the state |2>_(i) used for assistinga gate operation, a |2>_(i)-|e>_(i) transition being resonant with acavity mode of the cavity; a physical system Y_(j) (j=1, 2, . . . , N₂;N₂ being an integer not less than 1) provided in the cavity, thephysical system Y_(j) being different from the physical systems X_(i),the physical system Y_(j) comprising at least two energy statesincluding states |2>′_(j) and |e>′_(j), energy of the state |e>′_(j)being higher than energy of the state |2>′_(j), a |2>′_(j)−|e>′_(j)transition being resonant with the cavity mode; and a light source unitwhich applies laser beams to the cavity to manipulate a state of aphysical system X_(s) (where s is a natural number not more than N) anda state of physical system X_(t) (where t is a natural number not morethan N and different from s) of the physical systems X_(i), the laserbeams including a first laser beam resonating with a |1>_(s)-|e>_(s)transition, a second laser beam resonating with a |1>_(t)-|e>_(t)transition, and a third laser beam for collecting population in thestate |2>′_(j) in the |2>′_(j)-|e>′_(j) transition of the physicalsystem Y_(j).
 2. The quantum computer according to claim 1, wherein thelight source unit comprises: a light source which emits a laser beam;two beam splitters which divide the laser beam into a first branch laserbeam, a second branch laser beam and a third branch laser beam; a firstoptical modulator which modulates the first branch laser beam togenerate the first laser beam; a second optical modulator whichmodulates the second branch laser beam to generate the second laserbeam; and a third optical modulator which modulates the third branchlaser beam to generate the third laser beam.
 3. The quantum computeraccording to claim 1, wherein the light source unit comprises: a firstlight source; a first optical modulator which modulates a laser beamemitted from the first light source to generate the first laser beam; asecond light source; a second optical modulator which modulates a laserbeam emitted from the second light source to generate the second laserbeam; a third light source; and a third optical modulator whichmodulates a laser beam emitted from the third light source to generatethe third laser beam.
 4. The quantum computer according to claim 1,wherein the cavity comprises a first space region including the physicalsystems X_(i), and a second space region including the physical systemY_(i), the second space region being different from the first spaceregion, and the light source unit applies the third laser beam to thesecond space region.
 5. A quantum computing method comprising: preparingphysical systems X_(i) (i=1, 2, . . . , N; N being an integer not lessthan 2) provided in a cavity, each of the physical systems X_(i)comprising at least four states including states |0>_(i), |1>_(i),|2>_(i) and |e>_(i), energy of the state |e>_(i) is higher than energyof the states |0>_(i) and |1>i used for a qubit and energy of the state|2>_(i) used for assisting a gate operation, a |2>_(i)-|e>_(i)transition being resonant with a cavity mode of the cavity; preparing aphysical system Y_(j) (j=1, 2, . . . , N₂; N₂ being an integer not lessthan 1) provided in the cavity, the physical system Y_(j) beingdifferent from the physical systems X_(i), physical system Y_(j)comprising at least two energy states including |2>′_(j) and |e>′_(j),energy of the state |e>′_(j) being higher than energy of the state|2>′_(j), a |2>′_(j)-|e>′_(j) transition being resonant with the cavitymode; and applying laser beams to the cavity to manipulate state of aphysical system X_(s) (where s is a natural number not more than N) anda state of physical system X_(t) (where t is a natural number not morethan N and different from s) of the physical systems X_(i), the laserbeams including a first laser beam resonating with a |1>_(s)-|e>_(s)transition, a second laser beam resonating with a |1>_(t)-|e>_(t)transition, and a third laser beam for collecting population in thestate |2>′_(j) in the |2>′_(j)-|e>′_(j) transition of the physicalsystem Y_(j).
 6. The quantum computing method according to claim 5,further comprising: applying the first laser beam and the second laserbeam to the cavity, after applying the third laser beam to collectpopulation in the state |2>′_(j) of the physical system Y_(j).
 7. Thequantum computing method according to claim 5, wherein if the physicalsystems X_(i) comprises physical systems X_(i), X₂, . . . , X_(N) inascending order of frequency of a |1>_(i)-|e>_(i) transition, afrequency difference between a |1>_(i)-|e>₁ transition and a |1>₂-|e>₂transition is Δ, and a frequency difference between a |l>₁-|e>₁transition and a |1>_(k)-|e>_(k) transition is Δ_(k), then the pluralityof physical systems X_(i) satisfy: Ω₀ <Δ<N ₂ g/N and Ω ₀<Δ_(k) <N ₂ g/N,where g is a coupling constant between the cavity mode and each of thephysical systems X_(i), and Ω₀ is a parameter for determining a maximumvalue of Rabi frequencies of the first laser beam and the second laserbeam.
 8. The quantum computing method according to claim 5, whereinN₂≥N² is satisfied.